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The yield to maturity (YTM) is a critical concept when assessing the overall returns of a bond investment. It represents the annualized rate of return that an investor will receive if the bond is held until maturity and all payments are made as scheduled. YTM takes into account the coupon payments, the face value of the bond, the price paid for the bond, and the time remaining until maturity.

In our example, Bond X has a YTM of 7% while Bond Y has a YTM of 9%. These rates are insightful for investors since they provide a way to compare bonds with different coupon rates and prices. A higher YTM generally means a higher return, but also indicates a higher risk as it often correlates with lower bond prices.

The coupon rate is the annual interest rate paid by the bond issuer to the bondholder. It's expressed as a percentage of the bond's face value. For instance, Bond X with a face value of \(1,000 pays a coupon rate of 9%, which results in annual payments of \)90. Conversely, Bond Y, also with a face value of \(1,000, pays a lower coupon rate of 7%, resulting in annual payments of \)70.

The coupon rate does not change over the life of the bond, which means investors can rely on a stable income from these payments. However, the attractiveness of the coupon rate can vary with changes in market interest rates. If new bonds are issued with higher rates, existing bonds with lower coupon rates become less appealing, affecting their market price.

Time to maturity refers to the remaining time until a bond's principal is repaid. It is also part of the equation that determines a bond's price and yield. In our example, both Bond X and Bond Y have the same initial time to maturity of 13 years. As time passes and they move closer to their maturity dates, the bond prices will converge towards their face values, assuming all other factors remain constant.

This concept affects both the yield to maturity and the bond's price volatility. Generally, bonds with a longer time to maturity are more sensitive to interest rate changes, which means their prices can experience more significant fluctuations.

The bond pricing formula is a mathematical equation used to calculate the present value of a bond based on its future cash flows, which are the periodic coupon payments and the face value that will be paid at maturity. The formula considers the coupon rate, face value, yield to maturity, and time to maturity.

In the case of our two bonds, these variables will be substituted into the bond pricing formula to determine the current value of each bond. As we have seen, Bond X, with its higher coupon rate, will have a different price trajectory compared to Bond Y, with its lower coupon rate and higher yield to maturity.

A discount bond is sold for less than its face value. This typically happens when the bond's coupon rate is lower than the current market interest rates or yield to maturity. Bond Y, with a coupon rate of 7% and a YTM of 9%, is an example of this. Since the return that Bond Y offers is lower than the market rate, it must be sold at a discount to be an attractive investment option.

The price of a discount bond is expected to increase as it nears its maturity date, where it will be redeemed at its face value, assuming no default occurs. This price increase over time is known as 'accreting' and it partially compensates for the lower coupon payments received over the bond's life.

Conversely, a premium bond is sold for more than its face value. This occurs when the bond's coupon rate is higher than the prevailing market interest rates or its yield to maturity. Bond X, with a 9% coupon and a 7% YTM, fits this category. Investors are willing to pay more for this bond because the coupon payments are more generous than what is currently available in the market.

As the bond moves closer to its maturity date, its price will generally decrease, or 'amortize,' towards its face value. This gradual decrease is the opposite of the accretion process seen in discount bonds, reflecting the premium paid for higher income streams.